Arithmetic differential operators on a semistable model of ${\mathbb P}^1$

TL;DR

This paper investigates sheaves of logarithmic arithmetic differential operators on a semistable model of the projective line, revealing non-torsion properties of their first cohomology and analyzing the structure of associated graded sheaves.

Contribution

It provides the first explicit description of the associated graded sheaf of level zero differential operators and links cohomology classes to the sheaf's filtration structure.

Findings

First cohomology group of sheaves is non-torsion.

Explicit determination of the associated graded sheaf.

Connection between cohomology classes and the filtration structure.

Abstract

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one.